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Derivatives of Sine

Extracts from this document...

Introduction

Garner

Mark Andrew Garner

Judy Land

Math Standard Level

April 30, 2008

Derivatives of Sine Functions

Method

  1.  The following information was my investigation to find the derivative of the functionimage04.png.
  1. The following is a graph for the function image04.pngforimage12.png.

image25.png

TI – 83 Graph Window

Xmin =

-image07.png

Xmax =

image07.png

Xscl =

image08.png

Ymin =

-5

Ymax =

5

Yscl =

1


  1. The table below describes the behavior of the gradient of image04.pngforimage12.png.

Gradient behavior in given points of image04.png

x

Y

Gradient (+, image05.png, or 0)

image13.png

0

+

image05.pngimage20.png

1

0

image05.pngimage21.png

0

image05.png

image22.png

image05.png1

0

0

0

+

image08.png

1

0

image21.png

0

image05.png

image20.png

image05.png1

0

image07.png

0

+

  1. Because the derivative of a function is the gradient at a given point, my conjecture for image23.png
...read more.

Middle

image20.png

0

image07.png

1

image00.png

Relationship of dy/dx of f(x)=sinx versus Y-value of f(x)=cosx at a given X-value

X-Value

dy/dx f(x)=sinx

f(x)=cosx

image13.png

1

1

image05.pngimage20.png

0

0

image05.pngimage21.png

image05.png1

image05.png1

image22.png

0

0

0

1

1

image08.png

0

0

image21.png

image05.png1

image05.png1

image20.png

0

0

image07.png

1

1

image26.png


TI – 83 Graph Window

Xmin =

-image07.png

Xmax =

image07.png

Xscl =

image08.png

Ymin =

-5

Ymax =

5

Yscl =

1

The tables on the preceding page show that the derivative of image04.pngat the following plotted points image12.pngin increments of image08.png starting left to right x =image13.png. image27.png

In the following graph, the points are connected to form image23.png,  image16.png.

TI – 83 Graph Window

Xmin =

-image07.png

Xmax =

image07.png

Xscl =

image08.png

Ymin =

-5

Ymax =

5

Yscl =

1

image28.pngimage01.png


  1. The following is my investigation of the derivatives of functions in the formimage29.png. The red dots in the following graphs below represent the derivative of the function at a particular X-value.

image30.pngimage31.png

Concluded from the graphs above, my conjecture for the derivative of the functionimage29.png, image12.png

...read more.

Conclusion

image10.pngcan be written asimage11.png. Consider the following graphs where the red line representsimage10.png,image12.png, the gray line represents image11.png, and the red points represent the derivative of image10.pngat an X-value.

TI – 83 Graph Window

Xmin =

-image07.png

Xmax =

image07.png

Xscl =

image08.png

Ymin =

-5

Ymax =

5

Yscl =

1

image14.png

Because of the chain rule:image15.png andimage16.png,

Thus:

image10.pngorimage17.png

         =image18.png

image11.png

In conclusion, from my investigation of the Derivative of sine functions, I have discovered the following:

  • image16.png
  • image06.png, where the derivative is affected by
  1. a the amplitude
  2. b the frequency
  3. c the phase shift
  • The derivative of image10.pngcan be written asimage11.png.

*In this investigation, there are limitations to graphs given as examples. The values substituted only occurred image19.pngand, therefore, no negative values can be accounted for in the conjectures. Also, the graphs used only fall under image12.pngand , and, thus, cannot confirm the conjecture outside of these limits.

...read more.

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